Heights and Distances

high importance~2 Q in Tier 117 formulas⚡ 10 shortcuts5 subtopics

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Angles of elevation and depression

Master relation

\tan\theta=\frac{\text{height above eye level}}{\text{horizontal distance}}

Slant line

\sin\theta=\frac{h}{L},\quad \cos\theta=\frac{d}{L}\quad (L=\text{line-of-sight or ladder length})

Depression = elevation

\text{angle of depression at observer}=\text{angle of elevation at object}

45 degrees

\theta=45^\circ\Rightarrow d=h

30 degrees

\theta=30^\circ\Rightarrow d=\sqrt3\,h\ \Leftrightarrow\ h=\frac{d}{\sqrt3}

60 degrees

\theta=60^\circ\Rightarrow d=\frac{h}{\sqrt3}\ \Leftrightarrow\ h=\sqrt3\,d

Ladder (hypotenuse given)

h=L\sin\theta,\quad d=L\cos\theta

Same side

h=\frac{d}{\cot\alpha-\cot\beta}

Opposite sides

h=\frac{d}{\cot\alpha+\cot\beta}

30-60 pair

\cot30^\circ-\cot60^\circ=\frac{2}{\sqrt3}\Rightarrow h=\frac{d\sqrt3}{2}

Gap from a height

\text{gap}=h(\cot\alpha-\cot\beta)

Distance walked

\text{distance}=\text{speed}\times\text{time}

Chain to height

v\,t=h(\cot\theta_1-\cot\theta_2)

Statue on pedestal

\text{statue}=d(\tan\beta-\tan\alpha)

Tower on building

\text{tower}=d\tan\theta_{\text{top}}-\text{building}

Broken tree

\text{tree}=\frac{b}{\cos\theta}+b\tan\theta

Building + tower from one height

\text{tower}=\text{building}+d(\tan\beta-\tan\alpha)